Consider $\mathbb{R^{n+1}}$ as a smooth manifold. In some post I found on stackexchange it is claimed that $$\omega = \sum_{j=1}^{n+1}(-1)^{j-1}x_jdx_1\wedge\dots\wedge \hat{dx_j}\wedge\dots\wedge dx_{n+1}$$ is a volume form on $\mathbb{R}^{n+1}$. And it made me realize I do not really understand them.
A volume form on a smooth manifold $M$ of dimension $m$ is an $m$-form such that $M(p)\neq 0$ for all $p\in M$.
First off, in writing the ($n+1$)-form $\omega$ as above, have we implicitely chosen a chart on $\mathbb{R}^{n+1}$?
Secondly, why is this a volume form, if we evaluate it at the point $(0,\ldots,0)$ wouldn't we get zero? Or is this simply not a volume form in the definition I used?
Also, I do not really understand how this is evaluated. If we take for example $\omega(1,0,\ldots,0)$. Then I think it is equal to $dx_2\wedge\dots\wedge dx_{n+1}$. Which is a function from $(T_{(1,0,\ldots)}\mathbb{R}^{n+1})^{n+1}\cong \mathbb{R}^{n+1}$. But how do we take those elements and fill them in in the $dx_i$'s?
Finally could I get some more examples of volume forms on $\mathbb{R^{n+1}}$.
I think I understand the abstract definitions to a certain degree, but lack the knowledge and intuition to apply them to the real case.